Zepto.js is a minimalist JavaScript framework for mobile WebKit browsers, with a jQuery-compatible syntax.

Is it only recommended for use with apps that will be used on mobile devices or can it also be used in traditional computer browser web apps as well? What are the downsides of doing so?

I use jQuery a lot in my regular web apps (for computer browsers) and I've realized (after finding Zepto.js) that I don't always use all of jQuery's features, so Zepto.js is perfect for what I do.

Also, why is there focus on the 'WebKit' engine? Does it not work with mobile Internet Explorer on Windows Phone 7, Firefox Mobile on Android or Opera Mobile?

EDIT: The presence of $.os.android in addition to $.os.ios indicates that there is support for non-iOS platforms, so the last question is answered!

EDIT (final): I guess the real question is "Can the Zepto.js library, intended for use on mobile WebKit-based browsers, be used as a lighter-weight alternative to jQuery for the desktop also?" The answer is no.

Note: This answer is old, and not up to date anymore. Yes, Zepto started as a webkit-only, mobile-specific framework, but it has evolved since then as EBarr points out below.
So always check Zepto's website for the most up-to-date information.

Not to sound too harsh, but did you click the big button on the site to see the presentation?

You basically already answered the questions:

Yes, it's for mobile devices. Support for swipe and tap events doesn't exactly make sense on a desktop computer, where you use a mouse.

Yes, it's for WebKit specifically, because that's what's most used for mobile browsing. That means no support for IE/Firefox/Opera - mobile or desktop. As the presentation says, "No one is running IE6 on an iPhone". Most desktop-browser JS libraries exist specifically to eliminate browser differences. Zepto does the opposite.

jQuery is an offroad truck: Enormous but very capable both on- and off-road.
Zepto is a Formula-1 car: Small, fast, and built exclusively to be great on the racetrack, so it won't work right in any other situation.